List of top Questions asked in MHT CET

Evaluate \[ \int_{0}^{a} (a-x)^{\frac{3}{2}} x^2 \, dx \]

The area bounded by the curve \(y = x^3\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is

The equation of a plane passing through the intersection of the planes \(x + 2y - 3z + 2 = 0\) and \(6x + y + z + 1 = 0\) and parallel to the line \(x - 1 = y + 2 = 7 - z\) is

The angle between the lines \( \frac{x - 1}{4} = \frac{y - 3}{1} = \frac{z}{8} \) and \( \frac{x - 2}{2} = \frac{y + 1}{2} = \frac{z - 4}{1} \) is

The integrating factor of the differential equation \[ (1 + x^2) \, dt = \left( \tan^{-1} x - t \right) \, dx \text{ is} \]

If the p.m.f. is given by \( P(X) = k \left( \frac{4}{x} \right) \) for \( x = 0, 1, 2, 3, 4 \) and \( k>0 \), then the value of \( k \) is

The probability density function of a random variable \(X\) is given by \[ f(x) = \frac{k}{\sqrt{x}}, \quad 0 \le x \le 4 \] \[ f(x) = 0, \quad \text{otherwise} \] Find \(P(1<X<4)\).

The probability mass function of a random variable \(X\) is given by \[ P(X = x) = \frac{5cx}{2^5}, \quad x = 0,1,2,3,4,5 \] \[ P(X = x) = 0, \quad \text{otherwise} \] Then find \(P(X \le 2)\).

The dual of the statement pattern \( \sim p \land (q \lor t) \) is (where \( t \) is a tautology and \( c \) is a contradiction)

The statement pattern \[ [(p \vee q) \wedge \neg p] \wedge (\neg q) \] is

In a \( \triangle ABC \), if \( 2 \cos C = \sin B \cdot \csc A \), then

Radium decomposes at a rate proportional to the amount present. If half the original amount disappears in 1600 years, then the percentage loss in 100 years is

If the lines given by \( \frac{x-1}{2\lambda} = \frac{y-1}{5} = \frac{z-1}{2} \) and \( \frac{x+2}{\lambda} = \frac{y+3}{\lambda} = \frac{z+5}{1} \) are parallel, then the value of \( \lambda \) is

If \[ x \cos \theta + y \sin \theta = 5, \quad x \sin \theta - y \cos \theta = 3, \quad \text{then the value of} \, x^2 + y^2 \, \text{is} \]

The area of the region bounded by the parabola $x^2=16y$, the lines $y=1$, $y=4$ and the $y$-axis lying in the first quadrant is

Evaluate \[ \int_{-\pi/2}^{\pi/2} \sin^2 x\,dx \]

The area of the region bounded by the parabola \[ y^2 = 8x \quad \text{and its latus rectum is} \]

The displacement of a particle at time \(t\) is \[ s = t^3 - 4t^2 - 5t \] Then the velocity of the particle at \(t = 2\) sec is

If \[ A=\begin{bmatrix} 1 & 2 & i\\ 1 & 1 & 1\\ 1 & 1 & 0 \end{bmatrix}, \] then \( [\operatorname{adj}(\operatorname{adj}A)]^{-1} \) is

The eccentricity of the ellipse \[ y^2 + 4x^2 - 12x + 6y + 14 = 0 \] is

In a certain culture of bacteria, the rate of increase is proportional to the number present. It is found that the number doubles in 4 hours. Then the number of times the bacteria are increased in 12 hours is

Evaluate the integral \[ \int_{0}^{\pi/2} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}\, dx \]

Evaluate \( \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx \)

If one of the lines given by \( kx^2 + xy - y^2 = 0 \) bisects the angle between the co-ordinate axes, then values of \( k \) are:

The separate equations of the lines represented by the equation \( 3x^2 - 2\sqrt{3} xy - 3y^2 = 0 \) are